Positive hulls of random walks and bridges

TL;DR

This paper investigates the geometric properties of convex cones formed by positive hulls of random walks and bridges, providing explicit formulas for expected face counts and quermassintegrals using Stirling numbers.

Contribution

It introduces new formulas for expected geometric functionals of cones generated by random walks and bridges, linking them to Stirling numbers and their analogues.

Findings

Explicit expectations for face counts of cones

Formulas involving Stirling numbers and B-analogues

Insights into the geometry of random convex cones

Abstract

We study random convex cones defned as positive hulls of $d$-dimensional random walks and bridges. We compute expectations of various geometric functionals of these cones such as the number of $k$-dimensional faces and the sums of conic quermassintegrals of their $k$-dimensional faces. These expectations are expressed in terms of Stirling numbers of both kinds and their $B$-analogues.